If Kaspersky PURE 3.0 is stuck updating (22%ish) from version 13.0.2.558b to 13.0.2.558c and the log shows the file is stuck downloading on "stpass.exe.ap" (16mb)
You are more than likely on a slow internet connection and your download is timing out.
To fix this issue, change your Update Source from the http default to ftp://ftp.kaspersky.com
This change could possibly work for other Kaspersky clients as there a variety of client updates listed in ftp://ftp.kaspersky.com